Langevin dynamcs MCMC



Randomly exploring the posterior space.

Annealed

Jolicoeur-Martineau et al. (2022); Song and Ermon (2020a); Song and Ermon (2020b)

Generative Modeling by Estimating Gradients of the Data Distribution | Yang Song

Incoming

See log concave distributions for a family of distributions where this works especially well. Rob Salomone explains this well; see Hodgkinson, Salomone, and Roosta (2019). Holden Lee, Andrej Risteski introduce the connection between log-concavity and convex optimisation.

\[ x_{t+\eta} = x_t - \eta \nabla f(x_t) + \sqrt{2\eta}\xi_t,\quad \xi_t\sim N(0,I). \]

Left-field, Max Raginsky, Sampling Using Diffusion Processes, from Langevin to Schrödinger:

the Langevin process gives only approximate samples from \(\mu\). I would like to discuss an alternative approach that uses diffusion processes to obtain exact samples in finite time. This approach is based on ideas that appeared in two papers from the 1930s by Erwin Schrödinger in the context of physics, and is now referred to as the Schrödinger bridge problem.

References

Brosse, Nicolas, Éric Moulines, and Alain Durmus. 2018. The Promises and Pitfalls of Stochastic Gradient Langevin Dynamics.” In Proceedings of the 32nd International Conference on Neural Information Processing Systems, 8278–88. NIPS’18. Red Hook, NY, USA: Curran Associates Inc.
Dalalyan, Arnak S. 2017. Further and Stronger Analogy Between Sampling and Optimization: Langevin Monte Carlo and Gradient Descent.” arXiv:1704.04752 [Math, Stat], April.
Durmus, Alain, and Eric Moulines. 2016. High-Dimensional Bayesian Inference via the Unadjusted Langevin Algorithm.” arXiv:1605.01559 [Math, Stat], May.
Garbuno-Inigo, Alfredo, Franca Hoffmann, Wuchen Li, and Andrew M. Stuart. 2020. Interacting Langevin Diffusions: Gradient Structure and Ensemble Kalman Sampler.” SIAM Journal on Applied Dynamical Systems 19 (1): 412–41.
Ge, Rong, Holden Lee, and Andrej Risteski. 2020. Simulated Tempering Langevin Monte Carlo II: An Improved Proof Using Soft Markov Chain Decomposition.” arXiv:1812.00793 [Cs, Math, Stat], September.
Girolami, Mark, and Ben Calderhead. 2011. Riemann Manifold Langevin and Hamiltonian Monte Carlo Methods.” Journal of the Royal Statistical Society: Series B (Statistical Methodology) 73 (2): 123–214.
Grenander, Ulf, and Michael I. Miller. 1994. Representations of Knowledge in Complex Systems.” Journal of the Royal Statistical Society: Series B (Methodological) 56 (4): 549–81.
Hodgkinson, Liam, Robert Salomone, and Fred Roosta. 2019. Implicit Langevin Algorithms for Sampling From Log-Concave Densities.” arXiv:1903.12322 [Cs, Stat], March.
Jolicoeur-Martineau, Alexia, Rémi Piché-Taillefer, Ioannis Mitliagkas, and Remi Tachet des Combes. 2022. Adversarial Score Matching and Improved Sampling for Image Generation.” In.
Norton, Richard A., and Colin Fox. 2016. Tuning of MCMC with Langevin, Hamiltonian, and Other Stochastic Autoregressive Proposals.” arXiv:1610.00781 [Math, Stat], October.
Parisi, G. 1981. Correlation Functions and Computer Simulations.” Nuclear Physics B 180 (3): 378–84.
Rásonyi, Miklós, and Kinga Tikosi. 2022. On the Stability of the Stochastic Gradient Langevin Algorithm with Dependent Data Stream.” Statistics & Probability Letters 182 (March): 109321.
Shang, Xiaocheng, Zhanxing Zhu, Benedict Leimkuhler, and Amos J Storkey. 2015. Covariance-Controlled Adaptive Langevin Thermostat for Large-Scale Bayesian Sampling.” In Advances in Neural Information Processing Systems. Vol. 28. NIPS’15. Curran Associates, Inc.
Song, Yang, and Stefano Ermon. 2020a. Generative Modeling by Estimating Gradients of the Data Distribution.” In Advances In Neural Information Processing Systems. arXiv.
———. 2020b. Improved Techniques for Training Score-Based Generative Models.” In Advances In Neural Information Processing Systems. arXiv.
Welling, Max, and Yee Whye Teh. 2011. Bayesian Learning via Stochastic Gradient Langevin Dynamics.” In Proceedings of the 28th International Conference on International Conference on Machine Learning, 681–88. ICML’11. Madison, WI, USA: Omnipress.
Xifara, T., C. Sherlock, S. Livingstone, S. Byrne, and M. Girolami. 2014. Langevin Diffusions and the Metropolis-Adjusted Langevin Algorithm.” Statistics & Probability Letters 91 (Supplement C): 14–19.

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